Electronic Journal of Probability

Nonlinear historical superprocess approximations for population models with past dependence

Sylvie Méléard and Viet Chi Tran

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We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages. We define an interacting historical particle process  describing the  genealogies of the living individuals; it takes values in the space of point measures  on an infinite dimensional càdlàg path space. This individual-based process can be approximated by  a nonlinear historical superprocess, under the assumptions of large populations, small individuals and allometric demographies. Because of the interactions, the branching property fails and we use martingale problems and fine couplings between our population and independent branching particles. Our convergence theorem is illustrated by two examples of current interest in biology. The first one relates the biodiversity history of a population and its phylogeny, while the second treats a spatial model where individuals compete through their past trajectories.

Article information

Electron. J. Probab. Volume 17 (2012), paper no. 47, 32 pp.

Accepted: 18 June 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J68: Superprocesses 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Nonlinear historical superprocess Genealogical interacting particle system Limit theorem Evolution models

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Méléard, Sylvie; Tran, Viet Chi. Nonlinear historical superprocess approximations for population models with past dependence. Electron. J. Probab. 17 (2012), paper no. 47, 32 pp. doi:10.1214/EJP.v17-2093. https://projecteuclid.org/euclid.ejp/1465062369

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